$\mathrm{x}=\cos {55}^{\circ }\mathrm{y}=\cos {65}^{\circ },\mathrm{z}=\cos {175}^{\circ }\text{ then }\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=$

$\begin{array}{l}\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=\cos {55}^{\circ }\cos {65}^{\circ }+\cos {65}^{\circ }\cos {175}^{\circ }+\cos {175}^{\circ }\cos {55}^{\circ }\\ =(1/2)\left[\cos {120}^{\circ }+\cos {10}^{\circ }+\cos {240}^{\circ }+\cos {110}^{\circ }+\cos {230}^{\circ }+\cos {120}^{\circ }\right]\\ =(1/2)\left[2\cos {120}^{\circ }+\cos {240}^{\circ }+\cos {10}^{\circ }+\cos {230}^{\circ }+\cos {110}^{\circ }\right]\\ =(1/2)\left[2(-1/2)+(-1/2)+2\cos {120}^{\circ }\cos {110}^{\circ }+\cos {110}^{\circ }\right]\\ =(1/2)\left[(-3/2)+2(-1/2)\cos {110}^{\circ }+\cos {110}^{\circ }\right]=(1/2)[-3/2]=-3/4\end{array}$